## Abstract

In this paper we consider a general class of linear positive processes of integral type. These operators act on functions defined on unbounded interval. Among the particular cases included are Durrmeyer–Jain operators, Păltănea–Szász–Mirakjan operators and operators using Baskakov–Szász type bases. We focus on highlighting some approximation targeting different classes of functions. The main working tools are Bohman–Korovkin theorem, weighted *K*-functionals and moduli of smoothness.

## Authors

**Octavian Agratini
**Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Romania

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

## Keywords

Positive linear operator; Weighted *K*-functional; Modulus of smoothness; Durrmeyer–Jain operator

requires subscription: https://doi.org/10.1007/s11117-020-00752-y

## Cite this paper as:

O. Agratini, *Approximation properties of a family of integral type operators, *Positivity, **25** (2021), 97–108, https://doi.org/10.1007/s11117-020-00752-y

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# Approximation properties

of a family of integral type operators

###### Abstract.

In this paper we consider a general class of linear positive processes of integral type. These operators act on functions defined on unbounded interval. Among the particular cases included are Durrmeyer-Jain operators, Păltănea-Szász-Mirakjan operators and operators using Baskakov-Szász type bases. We focus on highlighting some approximation targeting different classes of functions. The main working tools are Bohman-Korovkin theorem, weighted $K$-functionals and moduli of smoothness.

Keywords and phrases: Positive linear operator, weighted $K$-functional, modulus of smoothness, Durrmeyer-Jain operator.

Mathematics Subject Classification: 41A36, 41A25.

## 1. Introduction

On the last five decades the interest of the study of positive approximation processes have emerged with growing evidence. In what follows we consider linear positive operators which approximate real valued functions defined on the unbounded interval ${\mathbb{R}}_{+}=[0,\mathrm{\infty})$. Among them, a special attention has been paid to operators which reproduce affine functions. Due to the linearity of the operators, this property is implied by the preservation of the test functions ${e}_{0}$ and ${e}_{1}$. Set ${e}_{j}$, $j\in {\mathbb{N}}_{0}=\{0\}\cup \mathbb{N}$, the monomial of degree $j$. The starting point is given by discrete operators of the following form

$$({L}_{n}f)(x)=\sum _{k=0}^{\mathrm{\infty}}{\lambda}_{n,k}(x)f({x}_{n,k}),n\in \mathbb{N},x\in {\mathbb{R}}_{+},$$ | (1.1) |

verifying the conditions

$$\sum _{k=0}^{\mathrm{\infty}}{\lambda}_{n,k}={e}_{0},$$ | (1.2) |

$$\sum _{k=0}^{\mathrm{\infty}}{x}_{n,k}{\lambda}_{n,k}={e}_{1},$$ | (1.3) |

where ${\lambda}_{n,k}$ is a continuous function defined on ${\mathbb{R}}_{+}$ with non-negative real values for each $k\in {\mathbb{N}}_{0}$, ${({x}_{n,k})}_{k\ge 0}$ represents a net on ${\mathbb{R}}_{+}$ $$ and function $f$ is chosen such that the series from relation (1.1) is convergent. Because such operators are not suitable for approximating discontinuous functions, they are generalized to integral type operators. One of the usual techniques is known as Durrmeyer method [12] which leads us to an approximation process, say ${({L}_{n}^{\ast})}_{n\ge 1}$, in spaces of integrable functions.

This type of integral generalization, studied in depth for the first time in 1981 by Derriennic [7], has been constant in the attention of many researches, among the most recent papers is that of Abel and Karsli [1].

For each $n\in \mathbb{N}$, let ${({\mu}_{n,k})}_{k\ge 1}$ be a sequence of continuous and positive functions defined on ${\mathbb{R}}_{+}$ such that the following relation is fulfilled

$${\int}_{0}^{\mathrm{\infty}}{\mu}_{n,k}(t)\mathit{d}t=1,k\in \mathbb{N}.$$ | (1.4) |

Setting ${I}_{n,k}(f)={\displaystyle {\int}_{{\mathbb{R}}_{+}}}{\mu}_{n,k}(t)f(t)\mathit{d}t$, $k\in \mathbb{N}$, we introduce the operators

$$({L}_{n}^{\ast}f)(x)={\lambda}_{n,0}(x)f(0)+\sum _{k=1}^{\mathrm{\infty}}{\lambda}_{n,k}(x){I}_{n,k}(f),$$ | (1.5) |

where $f\in \mathcal{F}({\mathbb{R}}_{+})$, this space consisting of all real valued functions $f$ defined on ${\mathbb{R}}_{+}$ with the property ${\lambda}_{n,k}f$ belongs to the Lebesgue space ${L}_{1}({\mathbb{R}}_{+})$ for each $k\in \mathbb{N}$ and the series from the right hand side of relation (1.5) is convergent. Denoting by ${C}_{B}({\mathbb{R}}_{+})$ the space of all continuous and bounded functions defined on ${\mathbb{R}}_{+}$, it is clear that ${C}_{B}({\mathbb{R}}_{+})\subset \mathcal{F}({\mathbb{R}}_{+})$.

###### Remark 1.1.

We mention that identity (1.4) is not an essential requirement. It is only important that the integral exists and is finite. If

$${\int}_{0}^{\mathrm{\infty}}{\mu}_{n,k}(t)\mathit{d}t={c}_{n,k},$$ |

where ${c}_{n,k}\in {\mathbb{R}}_{+}^{\ast}$, $k\in \mathbb{N}$, then we can define

$${I}_{n,k}(f)={c}_{n,k}^{-1}{\int}_{0}^{\mathrm{\infty}}{\mu}_{n,k}(t)f(t)\mathit{d}t,$$ |

and, this way, we reach the same operators ${L}_{n}^{\ast}$ given by (1.5).

It is obvious that ${L}_{n}^{\ast}$, $n\in \mathbb{N}$, operators are linear and positive. Moreover, relations (1.4) and (1.2) imply

$${L}_{n}^{\ast}{e}_{0}={e}_{0},n\in \mathbb{N}.$$ | (1.6) |

This also gives a formula for the norm of the operators, $\Vert {L}_{n}^{\ast}\Vert =1$.

To motivate the consistency of our study, our first concern is to highlight families of such operators that have already been studied. Next, some approximation properties of our general operators are revealed.

We point out that, alternatively, by using a two-dimensional kernel, the operators can be rewritten as follows

$$({L}_{n}^{\ast}f)(x)={\int}_{0}^{\mathrm{\infty}}{K}_{n}(x,t)f(t)\mathit{d}t,$$ |

where

$${K}_{n}(x,t)={\lambda}_{n,0}(x){\delta}_{x}(t)+\sum _{k=1}^{\mathrm{\infty}}{\lambda}_{n,k}(x){\mu}_{n,k}(t),$$ |

${\delta}_{x}(t)$ being Dirac generalized function.

Finally we introduce the $j$-th central moment of ${L}_{n}^{\ast}$ operators $(j\in {\mathbb{N}}_{0})$, i.e.,

$${\mathcal{M}}_{j}({L}_{n}^{\ast};x)=({L}_{n}^{\ast}{\phi}_{x}^{j})(x),\text{where}{\phi}_{x}(t)=t-x,(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}.$$ | (1.7) |

This can be accomplished by assuming that ${e}_{j}\in \mathcal{F}({\mathbb{R}}_{+})$.

## 2. Examples of particular cases of the operators ${L}_{n}^{\ast}$

The section includes three examples of known operators defined following the structure given by the relation (1.5).

1. Operators using Baskakov-Szász type bases. Choosing in (1.5)

$${\lambda}_{n,k}(x)=\left(\genfrac{}{}{0pt}{}{n+k-1}{k}\right){x}^{k}{(1+x)}^{-n-k},x\in [0,\mathrm{\infty}),k\in {\mathbb{N}}_{0},$$ |

and

$${\mu}_{n,k}(t)=\frac{n}{k!}{e}^{-nt}{(nt)}^{k},t\in [0,\mathrm{\infty}),k\in \mathbb{N},$$ |

the resulting operators (denoted by ${M}_{n}$) have been introduced in 2001 by Purshottam Agrawal and Ali Mohammad [4, Eq. (1.1)] in order to approximate a class of continuous functions of exponential growth. Their study was deepened by both the authors mentioned above [5] and by Gupta V., Gupta M.K. [13].

In this case, the conditions (1.2) and (1.4) are verified. Also, considering the network ${x}_{n,k}=k/n$, $k\in {\mathbb{N}}_{0}$, the relation (1.3) is fulfilled.

2. Szász-Mirakjan-Păltănea operators. The integral version we referred appeared in [18].

It depends on two parameters $\alpha >0$, $\rho >0$. In this case we have

${\lambda}_{\alpha ,k}(x)$ | $:={s}_{\alpha ,k}(x)={e}^{-\alpha x}{\displaystyle \frac{{(\alpha x)}^{k}}{k!}},x\in [0,\mathrm{\infty}),k\in {\mathbb{N}}_{0},$ | ||

${\mu}_{\alpha ,k}(x)$ | $:={\theta}_{\alpha ,k}^{\rho}(t)={\displaystyle \frac{\alpha {\rho}^{k\rho}}{\mathrm{\Gamma}(k\rho )}}{e}^{-\alpha \rho t}{(\alpha t)}^{k\rho -1},t\in [0,\mathrm{\infty}),k\in \mathbb{N},$ |

and the class of operators is designed as follows

$$({L}_{\alpha}^{\rho}f)(x)={e}^{-\alpha x}f(0)+\sum _{k=1}^{\mathrm{\infty}}{s}_{\alpha ,k}(x){\int}_{0}^{\mathrm{\infty}}{\theta}_{\alpha ,k}^{\rho}(t)f(t)\mathit{d}t.$$ |

Both relations (1.2), (1.4) and relation (1.3) with ${x}_{n,k}=k/n$, $k\in {\mathbb{N}}_{0}$, are satisfied. Păltănea continued the study of these operators in [19]. Among the most recent papers that deepen the study of ${({L}_{\alpha}^{\rho})}_{\alpha >0}$ we mention [17].

3. Durrmeyer-Jain operators. Using a Poisson-type distribution with two parameters given by

$${w}_{\beta}(k;\alpha )=\frac{\alpha}{k!}{(\alpha +k\beta )}^{k-1}{e}^{-(\alpha +k\beta )},k\in {\mathbb{N}}_{0},$$ |

for $\alpha >0$ and $\beta \in [0,1)$, Jain [14] introduced the following class of positive linear operators

$$({P}_{n}^{[\beta ]}f)(x)=\sum _{k=0}^{\mathrm{\infty}}{w}_{\beta}(k;nx)f\left(\frac{k}{n}\right),x\ge 0,$$ |

where $f\in C({\mathbb{R}}_{+})$ whenever the above series is convergent. In [2] an extension in Durrmeyer sense of ${P}_{n}^{[\beta ]}$ has been introduced, which was named ${\mathrm{\Lambda}}_{n}^{[\beta ]}$. In this case, in (1.5) we choose as follows

$${\lambda}_{n,k}(x)={w}_{\beta}(k;nx),x\in [0,\mathrm{\infty}),k\in {\mathbb{N}}_{0},$$ | (2.1) |

$${\mu}_{n,k}(t)=\frac{n}{1-\beta}{e}^{-nt/(1-\beta )}{\left(\frac{nt}{1-\beta}\right)}^{k-1},t\in [0,\mathrm{\infty}),k\in \mathbb{N}.$$ | (2.2) |

Relations (1.2) and (1.4) take place but (1.3) fails. We get

$${P}_{n}^{[\beta ]}{e}_{1}={(1-\beta )}^{-1}{e}_{1}.$$ |

For the special case $\beta =0$, ${L}_{n}^{\ast}\equiv {\mathrm{\Lambda}}_{n}^{[0]}$, $n\in \mathbb{N}$, turn into Phillips operators [20]. Among the first papers in which the approximation properties of these operators were studied, we mention [16].

###### Remark 2.1.

The integral operators presented in Examples 2 and 3 enjoy the property of keeping the affine functions as fixed points. Relation (1.6) holds and it is easy to verify that ${L}_{\alpha}^{\rho}{e}_{1}={\mathrm{\Lambda}}_{n}^{[\beta ]}{e}_{1}={e}_{1}.$

Inspired by this fact, for certain results that we will establish in the next section, we will impose the following condition to be fulfilled

$$\sum _{k=1}^{\mathrm{\infty}}{\lambda}_{n,k}{I}_{n,k}({e}_{1})={e}_{1},n\in \mathbb{N}.$$ | (2.3) |

This additional condition ensures the identity

$${\mathcal{M}}_{2}({L}_{n}^{\ast};x)=({L}_{n}^{\ast}{e}_{2})(x)-{x}^{2},x\ge 0,$$ | (2.4) |

and the quantity is strictly positive for $x>0$.

###### Remark 2.2.

Regarding the central moment of the second order for the operators presented above, we have the following relations

$$\{\begin{array}{c}{\mathcal{M}}_{2}({M}_{n};x)=\frac{x(x+2)}{n},{\mathcal{M}}_{2}({L}_{\alpha}^{\rho};x)=\frac{\rho +1}{\alpha \rho}x,\hfill \\ {\mathcal{M}}_{2}({\mathrm{\Lambda}}_{n}^{[\beta ]};x)=\frac{1+{(1-\beta )}^{2}}{n(1-\beta )}x,\hfill \end{array}$$ | (2.5) |

see respectively [4, Lemma 1], [19, Eq. (2.1)], [2, Eq. (15)].

## 3. Results

###### Remark 3.1.

For each $n\in \mathbb{N}$, ${L}_{n}^{\ast}$ maps continuously ${C}_{B}({\mathbb{R}}_{+})$ into itself. Indeed, for $f\in {C}_{B}({\mathbb{R}}_{+})$ we get $|{I}_{n,k}(f)|\le \Vert f\Vert $, $k\in {\mathbb{N}}_{0}$, and based on relation (1.2), we deduce $\Vert {L}_{n}^{\ast}f\Vert \le \Vert f\Vert $, where $\parallel \cdot \parallel $ stands for the supremum norm on ${\mathbb{R}}_{+}$.

Following the line of Ditzian and Totik [10, §1.2], we consider $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}^{\ast}$ an admissible weight function. In order to give an estimate of the approximation error, we use the weighted $K$-functional of second order $f\in {C}_{B}({\mathbb{R}}_{+})$ defined as follows

$${K}_{2,\phi}(f,t):=\underset{g}{inf}\{\parallel f-g\parallel +t\parallel {\phi}^{2}{g}^{\prime \prime}\parallel :{g}^{\prime}\in A{C}_{loc}({\mathbb{R}}_{+})\},t>0,$$ | (3.1) |

where ${g}^{\prime}\in A{C}_{loc}({\mathbb{R}}_{+})$ means that $g$ is differentiable and ${g}^{\prime}$ is absolutely continuous on every compact of ${\mathbb{R}}_{+}^{\ast}$, [10, Eq. 2.1.1].

###### Theorem 3.2.

Proof. Let $x>0$ be fixed and $g:{\mathbb{R}}_{+}\to \mathbb{R}$ be twice differentiable such that ${g}^{\prime}\in A{C}_{loc}({\mathbb{R}}_{+})$. Starting from Taylor’s expansion

$$g(u)=g(x)+{g}^{\prime}(x)(u-x)+{\int}_{x}^{u}{g}^{\prime \prime}(t)(u-t)\mathit{d}t,u\ge 0,$$ |

and knowing that (2.3) holds, in other words ${L}_{n}^{\ast}$ reproduces linear functions, we have

$$({L}_{n}^{\ast}g)(x)-g(x)={L}_{n}^{\ast}({\int}_{x{e}_{0}}^{{e}_{1}}{g}^{\prime \prime}(t)({e}_{1}-t)\mathit{d}t,x).$$ |

Since ${\phi}^{2}$ is concave, for every $t=(1-\lambda )u+\lambda x$, $\lambda \in (0,1)$, we get

$${\phi}^{2}(t)\ge (1-\lambda ){\phi}^{2}(u)+\lambda {\phi}^{2}(x)\ge \lambda {\phi}^{2}(x)>0,$$ |

and consequently

$$\frac{|t-u|}{{\phi}^{2}(u)}=\frac{\lambda |x-u|}{{\phi}^{2}(t)}\le \frac{|x-u|}{{\phi}^{2}(x)}.$$ |

It turns out that

$\left|{\displaystyle {\int}_{x}^{u}}{g}^{\prime \prime}(u)(u-t)\mathit{d}t\right|$ | $\le \Vert {\phi}^{2}{g}^{\prime \prime}\Vert \left|{\displaystyle {\int}_{x}^{u}}{\displaystyle \frac{|t-u|}{{\phi}^{2}(t)}}\mathit{d}t\right|$ | ||

$\le \Vert {\phi}^{2}{g}^{\prime \prime}\Vert \left|{\displaystyle {\int}_{x}^{u}}{\displaystyle \frac{|x-u|}{{\phi}^{2}(x)}}\mathit{d}t\right|$ | |||

$=\Vert {\phi}^{2}{g}^{\prime \prime}\Vert {\displaystyle \frac{{(x-u)}^{2}}{{\phi}^{2}(x)}}.$ |

Applying the linear positive operator ${L}_{n}^{\ast}$ and taking into account (2.4), we get

$${L}_{n}^{\ast}({\int}_{x{e}_{0}}^{{e}_{1}}{g}^{\prime \prime}(\cdot )({e}_{1}-t)\mathit{d}t;x)\le \Vert {\phi}^{2}{g}^{\prime \prime}\Vert \frac{({L}_{n}^{\ast}{e}_{2})(x)-{x}^{2}}{{\phi}^{2}(x)},$$ |

and further

$|({L}_{n}^{\ast}f)(x)-f(x)|$ | $\le |{L}_{n}^{\ast}(f-g;x)|+|g(x)-f(x)|+|({L}_{n}^{\ast}g)(x)-g(x)|$ | ||

$\le 2\Vert f-g\Vert +\Vert {\phi}^{2}{g}^{\prime \prime}\Vert {\displaystyle \frac{({L}_{n}^{\ast}{e}_{2})(x)-{x}^{2}}{{\phi}^{2}(x)}}.$ |

In the above we used Remark 3.1. At this point, taking the infimum over all $g$ with ${g}^{\prime}\in A{C}_{loc}({\mathbb{R}}_{+})$, based on (3.1), we get the desired result. $\mathrm{\square}$

In accordance with [10, Theorem 2.1.1], for some constants $M>0$ and ${t}_{0}$, the following inequality

$$ |

takes place, where

$${\omega}_{2,\phi}{(f,t)}_{\mathrm{\infty}}=\underset{0\le h\le t}{sup}\underset{x\pm h\phi (x)\ge 0}{sup}|f(x-\phi (x)h)-2f(x)+f(x+\phi (x)h)|.$$ |

From Theorem 3.2 we deduce

###### Corollary 3.3.

Under the same assumptions of Theorem 3.2, the following inequality

$$|({L}_{n}^{\ast}f)(x)-f(x)|\le \stackrel{~}{M}{\omega}_{2,\phi}{(f;\frac{1}{\phi (x)}\sqrt{({L}_{n}^{\ast}{e}_{2})(x)-{x}^{2}})}_{\mathrm{\infty}}$$ |

holds, where $\stackrel{~}{M}>0$ is a certain constant.

###### Theorem 3.4.

Let $K\subset {\mathbb{R}}_{+}$ be a compact interval and ${L}_{n}^{\ast}$, $n\in \mathbb{N}$, be defined by (1.5) such that (2.3) takes place. If

$$\underset{n\to \mathrm{\infty}}{lim}{L}_{n}^{\ast}{e}_{2}={e}_{2}\text{uniformly on}K,$$ |

then

$$\underset{n\to \mathrm{\infty}}{lim}{L}_{n}^{\ast}f=f\text{uniformly on}K,$$ |

provided $f\in C({\mathbb{R}}_{+})\cap \mathcal{F}({\mathbb{R}}_{+})$.

For investigating other properties of our sequence, we need the following technical result.

###### Lemma 3.5.

Let ${L}_{n}^{\ast}$, $n\in \mathbb{N}$, be defined by (1.5). For any constant $\beta \in (0,1]$ and function $h:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, the following inequality

$$({L}_{n}^{\ast}{h}^{\beta})(x)\le {\lambda}_{n,0}(x){h}^{\beta}(0)+{(({L}_{n}^{\ast}{h}^{2})(x))}^{\beta /2},x\ge 0,$$ | (3.2) |

takes place.

Proof. We use Hölder’s inequality both for integrals and for sums. Considering $r=2{\beta}^{-1}$ in the relation $1/r+1/s=1$, $r>0$, $s>0$, we can write

${\int}_{0}^{\mathrm{\infty}}}{\mu}_{n,k}(t){h}^{\beta}(t)\mathit{d}t$ | $\le {\left({\displaystyle {\int}_{0}^{\mathrm{\infty}}}{\mu}_{n,k}(t)\mathit{d}t\right)}^{1/s}{\left({\displaystyle {\int}_{0}^{\mathrm{\infty}}}{\mu}_{n,k}(t){h}^{2}(t)\mathit{d}t\right)}^{1/r}$ | ||

$={I}_{n,k}^{\beta /2}({h}^{2}).$ |

Further,

$\sum _{k=1}^{\mathrm{\infty}}}{\lambda}_{n,k}(x){I}_{n,k}({h}^{\beta})$ | $\le {\left({\displaystyle \sum _{k=1}^{\mathrm{\infty}}}{\lambda}_{n,k}(x)\right)}^{1/s}{\left({\displaystyle \sum _{k=1}^{\mathrm{\infty}}}{\lambda}_{n,k}(x){I}_{n,k}({h}^{2})\right)}^{1/r}$ | ||

$={\left({\displaystyle \sum _{k=1}^{\mathrm{\infty}}}{\lambda}_{n,k}(x){I}_{n,k}({h}^{2})\right)}^{\beta /2}$ | |||

$={(({L}_{n}^{\ast}{h}^{2})(x)-{\lambda}_{n,0}(x){h}^{2}(0))}^{\beta /2}\le {(({L}_{n}^{\ast}{h}^{2})(x))}^{\beta /2}$ |

and the conclusion occurs. $\mathrm{\square}$

Next, we study the approximation properties for functions that are not necessarily bounded. We start with functions that satisfy a Hölder type condition with $\beta \in (0,1]$ on $E\subseteq {\mathbb{R}}_{+}$, more precisely functions $f:{\mathbb{R}}_{+}\to \mathbb{R}$ which verify the relation

$$|f(x)-f(y)|\le {C}_{\beta ,f}{|x-y|}^{\beta},(x,y)\in {\mathbb{R}}_{+}\times E,$$ | (3.3) |

where ${C}_{\beta ,f}$ is a positive constant depending only on $\beta $ and $f$. Clearly, such functions belong to $\mathcal{F}({\mathbb{R}}_{+})$. For $\beta >1$ relation (3.3) leads us to constant functions on $E$, so we are not interested at this time.

We prove a relation between the local smoothness of functions and the local approximation.

###### Theorem 3.6.

Let ${L}_{n}^{\ast}$, $n\in \mathbb{N}$, be defined by (1.5). If $f$ satisfies relation (3.3), then

$$|({L}_{n}^{\ast}f)(x)-f(x)|\le {C}_{\beta ,f}({\lambda}_{n,0}(x){x}^{\beta}+{\mathcal{M}}_{2}^{\beta /2}({L}_{n}^{\ast};x)+2{d}^{\beta}(x,E)),$$ | (3.4) |

$x\ge 0$, where ${\mathcal{M}}_{2}({L}_{n}^{\ast};x)$ is given at (1.7) and $d(x,E)$ represents the distance between $x$ and $E$.

Proof. It is clear that (3.3) holds for any $x\in {\mathbb{R}}_{+}$ and $y\in \overline{E}$, the closure of $E$ in $\mathbb{R}$. Let $(x,{x}_{0})\in {\mathbb{R}}_{+}\times \overline{E}$ such that

$$|x-{x}_{0}|=d(x,E):=inf\{|x-y|:y\in E\}.$$ |

Since $|f-f(x)|\le |f-f({x}_{0})|+|f({x}_{0})-f(x)|$ and ${L}_{n}^{\ast}$ is a linear positive operator satisfying (1.6), we can write

$|({L}_{n}^{\ast}f)(x)-f(x)|$ | $\le {L}_{n}^{\ast}(|f-f({x}_{0})|;x)+|f(x)-f({x}_{0})|$ | |||

$\le {L}_{n}^{\ast}({C}_{\beta ,f}{|{e}_{1}-{x}_{0}|}^{\beta};x)+{C}_{\beta ,f}{|x-{x}_{0}|}^{\beta}.$ | (3.5) |

On the other hand, choosing in (3.2) $h:=|{\phi}_{x}|$, see (1.7), we deduce

$${L}_{n}^{\ast}({|{e}_{1}-x|}^{\beta};x)\le {\lambda}_{n,0}(x){x}^{\beta}+{\mathcal{M}}_{2}^{\beta /2}({L}_{n}^{\ast};x).$$ | (3.6) |

Based on the elemental inequality ${|t-{x}_{0}|}^{\beta}\le {|t-x|}^{\beta}+{|x-{x}_{0}|}^{\beta}$, $$, on the monotonicity property of the operator ${L}_{n}^{\ast}$ as well as on relations (3.6) and (1.6), we get

$${L}_{n}^{\ast}({|{e}_{1}-{x}_{0}|}^{\beta};x)\le {\lambda}_{n,0}(x){x}^{\beta}+{\mathcal{M}}_{2}^{\beta /2}({L}_{n}^{\ast};x)+{d}^{\beta}(x,E).$$ |

Returning at (3), the proof is complete. $\mathrm{\square}$

###### Remark 3.7.

In relation (3.4), in the particular cases $x\in \overline{E}$ and $E={\mathbb{R}}_{+}^{\ast}$, the term $d(x,E)$ vanishes.

Further, we discuss about the type of convergence of the sequence ${({L}_{n}^{\ast}f)}_{n\ge 1}$ to the continuous function $f$ that it approximates. On compact intervals, if $f$ is continuous, the uniform convergence of ${({L}_{n}^{\ast}f)}_{n\ge 1}$ to $f$ takes place, see Theorem 3.4. On unbounded intervals, if $f$ is continuous, usually only pointwise convergence occurs. We will indicate sufficient conditions which ensure the uniform convergence on unbounded intervals $J\subseteq {\mathbb{R}}_{+}$. For discrete approximation processes such an approach can be found, for example, in [3]. Similar results have been obtained for a long time ago by Jésus de la Cal and Javier Cárcamo [6] for families of operators of probabilistic type over non-compact intervals. The authors used representation of the operators in terms of appropriate stochastic processes.

Set $UC({\mathbb{R}}_{+})$ the space of all uniformly continuous real valued function on ${\mathbb{R}}_{+}$. Let $\tau :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a continuous one-to-one function such that $\tau (0)=0$. We define the following two functions

$${f}^{\ast}=f\circ {\tau}^{-1},{\tau}_{x}(t)=|\tau (t)-\tau (x)|,t\ge 0,x\ge 0,$$ | (3.7) |

where $f\in {C}_{B}({\mathbb{R}}_{+})$.

###### Theorem 3.8.

Let ${L}_{n}^{\ast}$, $n\in \mathbb{N}$, be defined by (1.5) such that ${\lambda}_{n,0}(0)=1$ and let the functions ${\tau}_{x}$, ${f}^{\ast}$ be given by (3.7).

If $f\in UC({\mathbb{R}}_{+})$ and there are two sequence of positive real numbers, ${({\gamma}_{1,n})}_{n\ge 1}$, ${({\gamma}_{2,n})}_{n\ge 1}$ that converge to zero satisfying the conditions

$${\lambda}_{n,0}(x)(1+\tau (x))\le {\gamma}_{1,n},x\ge a,$$ | (3.8) |

$${L}_{n}^{\ast}({\tau}_{x};x)\le {\gamma}_{2,n},x\ge a,$$ | (3.9) |

for a certain $a\ge 0$, then ${({L}_{n}^{\ast}f)}_{n\ge 1}$ converges uniformly to $f$ on the interval $[a,\mathrm{\infty})$.

Proof. The main tool used to motivate our statement is modulus of smoothness associated to any bounded function $g$ defined on ${\mathbb{R}}_{+}$ expressed by the following formula

$$\omega (g;\delta )=sup\{|g({x}^{\prime})-g({x}^{\prime \prime})|:{x}^{\prime},{x}^{\prime \prime}\in {\mathbb{R}}_{+},|{x}^{\prime}-{x}^{\prime \prime}|\le \delta \},\delta \ge 0.$$ | (3.10) |

Its most prominent properties can be found, e.g., in [8, p. 40-44]. We recall three of them, useful for our proof. The function $\omega (g;\cdot )$ is nondecreasing,

$$\omega (g;\lambda \delta )\le (1+\lambda )\omega (g;\delta ),\delta \ge 0\text{and}\lambda \ge 0,$$ | (3.11) |

and, if in addition $g$ is uniformly continuous on ${\mathbb{R}}_{+}$, then

$$\underset{\delta \to {0}^{+}}{lim}\omega (g;\delta )=0.$$ | (3.12) |

We consider $x\ge a>0$ arbitrarily fixed. Based on identities (1.5) and (3.10) we get

$|({L}_{n}^{\ast}f)(x)-f(x)|$ | $=|{L}_{n}^{\ast}({f}^{\ast}\circ \tau ;x)-{f}^{\ast}(\tau (x))|$ | |||

$\le {L}_{n}^{\ast}(|{f}^{\ast}\circ \tau -{f}^{\ast}(\tau (x))|;x)$ | ||||

$={\lambda}_{n,0}(x)|{f}^{\ast}(\tau (0))-{f}^{\ast}(\tau (x))|$ | ||||

$+{\displaystyle \sum _{k=1}^{\mathrm{\infty}}}{\lambda}_{n,k}(x){I}_{n,k}({f}^{\ast}\circ \tau -{f}^{\ast}(\tau (x)))$ | ||||

$:={A}_{x}+{B}_{x}.$ | (3.13) |

At first we evaluate ${A}_{x}$. If $g$ is uniformly continuous function on ${\mathbb{R}}_{+}$, then its growth on domain is at most affine, i.e., there exist ${c}_{0},{c}_{1}$ positive constants such that $|g(t)|\le {c}_{1}t+{c}_{0}$ for all $t\in {\mathbb{R}}_{+}$, see, e.g., [9, p. 48, Problème 4] or [11]. Because ${f}^{\ast}\in UC({\mathbb{R}}_{+})$, in view of this result, we can write

$${A}_{x}\le {\lambda}_{n,0}(x)(|{f}^{\ast}(0)|+|{f}^{\ast}(\tau (x))|)\le \stackrel{~}{c}{\lambda}_{n,0}(1+\tau (x))\le \stackrel{~}{c}{\gamma}_{1,n},$$ | (3.14) |

where $\stackrel{~}{c}=\mathrm{max}\{2{c}_{0},{c}_{1}\}$. We also used relation (3.8).

Next we evaluate ${B}_{x}$. Clearly, $({L}_{n}^{\ast}{\tau}_{x})(x)>0$ for $x\ne 0$. The definition of modulus of smoothness and property (3.11) applied for

$$\lambda =|\tau (t)-\tau (x)|/({L}_{n}{\tau}_{x})(x)$$ |

allow us to write

$|{f}^{\ast}(\tau (t))-{f}^{\ast}(\tau (x))|$ | $\le \omega ({f}^{\ast};|\tau (t)-\tau (x)|)$ | ||

$\le \left(1+{\displaystyle \frac{{\tau}_{x}(t)}{{L}_{n}^{\ast}({\tau}_{x};x)}}\right)\omega ({f}^{\ast};{L}_{n}^{\ast}({\tau}_{x};x))$ | |||

$\le \left(1+{\displaystyle \frac{{\tau}_{x}(t)}{{L}_{n}^{\ast}({\tau}_{x};x)}}\right)\omega ({f}^{\ast};{\gamma}_{2,n}).$ |

In the above we also used the monotonicity of the function $\omega ({f}^{\ast};\cdot )$ as well as the hypothesis specified at (3.9). Since ${f}^{\ast}$ is bounded on the domain, consequently $\omega ({f}^{\ast};\cdot )$ is well defined. Thus, we have

$${B}_{x}\le \left({u}_{1}(x)+\frac{{u}_{2}(x)}{{L}_{n}^{\ast}({\tau}_{x};x)}\right)\omega ({f}^{\ast};{\gamma}_{2,n})\le 2\omega ({f}^{\ast};{\gamma}_{2,n}),$$ | (3.15) |

where

$$\{\begin{array}{ccc}0\le {u}_{1}(x):=1-{\lambda}_{n,0}(x)\le 1,\hfill & & \\ 0\le {u}_{2}(x):={L}_{n}^{\ast}({\tau}_{x},x)-{\tau}_{x}(0){\lambda}_{n,0}(x)\le {L}_{n}^{\ast}({\tau}_{x},x).\hfill & & \end{array}$$ |

Using the inequalities (3.14) and (3.15), relation (3) implies

$$|({L}_{n}^{\ast}f)(x)-f(x)|\le \stackrel{~}{c}{\gamma}_{1,n}+2\omega ({f}^{\ast};{\gamma}_{2,n}),x\ge a.$$ | (3.16) |

Relation ${\lambda}_{n,0}(0)=1$ corroborated with (1.2) implies ${\lambda}_{n,k}(0)=0$, $k\in \mathbb{N}$, consequently ${L}_{n}^{\ast}$ enjoys the interpolating property in $x=0$. Thus, relation (3.16) is true in the particular case $a=0$ as well.

Property (3.12) guarantees $\omega ({f}^{\ast};{\gamma}_{2,n})$ tends to zero as $n$ tends to infinity. Alongside (3.8), this statement leads to the completion of the proof. $\mathrm{\square}$

###### Example 3.9.

We apply Theorem 3.8 to Durrmeyer-Jain operators ${\mathrm{\Lambda}}_{n}^{[\beta ]}$, $n\ge 1$, where $\beta \in [0,1)$. Let us choose $a=1$ and $\tau (x)=\sqrt{x}$. Taking in view (2.1) and (2.2), we can write

$${w}_{\beta}(0;nx)\left(1+\sqrt{x}\right)\le \frac{1}{\sqrt{n}}:={\gamma}_{1,n},x\ge 1,$$ |

and

${\mathrm{\Lambda}}_{n}^{[\beta ]}({\tau}_{x};x)$ | $={w}_{\beta}(0;nx)\sqrt{x}+{\displaystyle \sum _{k=1}^{\mathrm{\infty}}}{w}_{\beta}(k;nx){\displaystyle {\int}_{0}^{\mathrm{\infty}}}{\mu}_{n,k}(t){\displaystyle \frac{|t-x|}{\sqrt{t}+\sqrt{x}}}\mathit{d}t$ | ||

$\le {\displaystyle \frac{1}{\sqrt{n}}}+{\displaystyle \frac{1}{\sqrt{x}}}{\mathcal{M}}_{2}^{1/2}({\mathrm{\Lambda}}_{n}^{[\beta ]};x)={\displaystyle \frac{1+\stackrel{~}{\beta}}{\sqrt{n}}},x\ge 1,$ |

where $\stackrel{~}{\beta}={\left({\displaystyle \frac{1+{(1-\beta )}^{2}}{1-\beta}}\right)}^{1/2}$, see (2.5).

The requirements of Theorem 3.8 being fulfilled, we deduce that ${({\mathrm{\Lambda}}_{n}^{[\beta ]}f)}_{n\ge 1}$ converges uniformly to $f$ on $[1,\mathrm{\infty})$ for any $f\in UC({\mathbb{R}}_{+})$.

Concluding remark. The results presented in this section have a unifying character in the sense that they can be applied to several particular classes of operators. As a weak point we admit that these results are not as spectacular as the ones that could be obtained for each particular case.

## References

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- [2] Agratini, O., On an approximation process of integral type, Appl. Math. Comp., 236(2014), 195-201.
- [3] Agratini, O., Uniform approximation of some classes of linear positive operators expressed by series, Applicable Analysis, 94(2015), No. 8, 1662-1669.
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- [7] Derriennic, M.M., Sur l’approximation de fonctions intégrables sur $[0,1]$ par des polynômes de Bernstein modifiés, J. Approx. Theory, 31(1981), 325-343.
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- [13] Gupta, V., Gupta, M.K., Rate of convergence for certain families of summation-integral type operators, J. Math. Anal. Appl., 296(2004), 608-618.
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[1] Abel, U., Karsli, H., *Asymptotic expansions for Bernstein–Durrmeyer–Chlodovsky polynomials.* Results Math. **73**, Article 104 (2018)

[2] Agratini, O., *On an approximation process of integral type.* Appl. Math. Comput. **236**, 195–201 (2014), MathSciNet MATH Google Scholar

[3] Agratini, O., *Uniform approximation of some classes of linear positive operators expressed by series*. Appl. Anal. **94**(8), 1662–1669 (2015), MathSciNet Article Google Scholar

[4] Agrawal, P.N., *Mohamed, A.J.: Linear combination of a new sequence of linear positive operators*. Rev. Un. Mat. Argentina **42**(2), 57–65 (2001), MathSciNet MATH Google Scholar

[5] Agrawal, P.N., *Mohamed, A.J.: On L_{p} *-approximation by a linear combination of a new sequence of linear positive operators. Turk. J. Math. **27**, 389–405 (2003), Google Scholar

[6] de la Cal, J., Cárcamo, J., *On uniform approximation by some classical Bernstein-type operators*. J. Math. Anal. Appl. **279**(2), 625–638 (2003), MathSciNet Article Google Scholar

[7] Derriennic, M.M., *Sur l’approximation de fonctions intégrables sur [0,1] *par des polynômes de Bernstein modifiés. J. Approx. Theory **31**, 325–343 (1981), MathSciNet Article Google Scholar

[8] De Vore, R.A., Lorentz, G.G.,* Constructive Approximation. A Series of Comprehensive Studies in Mathematics*, vol. 303. Springer, Berlin (1993), Google Scholar

[9] Dieudonné, J., *Éléments d’Analyse. Tome 1: Fondements de l’Analyse Moderne*. Gauthiers Villars, Paris (1968), MATH Google Scholar

[10] Ditzian, Z., Totik, V., *Moduli of Smoothness. Springer Series in Computational Mathematics*, vol. 9. Springer-Verlag, New York Inc., New York (1987), MATH Google Scholar

[11] Djebali, S., *Uniform continuity and growth of real continuous functions.* Int. J. Math. Educ. Sci. Technol. **32**(5), 677–689 (2001),MathSciNet Article Google Scholar

[12] Durrmeyer, J.-L., *Une formule d’inversion de la transformée de Laplace: applications à la théorie des moments*. Thèse de 3e cycle, Faculté des Sciences de l’Université de Paris (1967)

[13] Gupta, V., Gupta, M.K., *Rate of convergence for certain families of summation-integral type operators*. J. Math. Anal. Appl. **296**, 608–618 (2004), MathSciNet Article Google Scholar

[14] Jain, G.C., *Approximation of functions by a new class of linear operators*. J. Aust. Math. Soc. **13**(3), 271–276 (1972), MathSciNet Article Google Scholar

[15] Korovkin, P.P., *On convergence of linear positive operators in the space of continuous functions*. Dokl. Akad. Nauk SSSR (N.S.) **90**, 961–964 (1953). (Russian), MathSciNet Google Scholar

[16] May, C.P., *On Phillips operators. J. Approx*. Theory **20**(4), 315–332 (1977), MathSciNet Article Google Scholar

[17] Mursaleen, M., Rahman, S., Ansari, K.J., *On the approximation by Bézier-Păltănea operators based on Gould-Hopper polynomials*. Math. Commun. **24**(2), 147–164 (2019), MathSciNet MATH Google Scholar

[18] Păltănea, R., *Modified Szász-Mirakjan operators of integral form*. Carpathian J. Math. **24**(3), 378–385 (2008), MATH Google Scholar

[19] Păltănea, R., *Simultaneous approximation by a class of Szász-Mirakjan operators*. J. Appl. Funct. Anal. **9**(3–4), 356–368 (2014), MathSciNet MATH Google Scholar

[20] Phillips, R.S., *An inversion formula for Laplace transforms and semigroups of linear operators*. Ann. Math. Second Ser. **59**, 325–356 (1954), Article Google Scholar